1,560 research outputs found

    A Unifying Theory for Graph Transformation

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    The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Quantum ergodicity on the Bruhat-Tits building for PGL(3,F)\text{PGL}(3, F) in the Benjamini-Schramm limit

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    We study eigenfunctions of the spherical Hecke algebra acting on L2(Γn\G/K)L^2(\Gamma_n \backslash G / K) where G=PGL(3,F)G = \text{PGL}(3, F) with FF a non-archimedean local field of characteristic zero, K=PGL(3,O)K = \text{PGL}(3, \mathcal{O}) with O\mathcal{O} the ring of integers of FF, and (Γn)(\Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

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    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on nn-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth dd and using kk labels, we can solve - Independent Set in time 2O(dk)nO(1)2^{O(dk)}\cdot n^{O(1)} using O(dk2logn)O(dk^2\log n) space; - Max Cut in time nO(dk)n^{O(dk)} using O(dklogn)O(dk\log n) space; and - Dominating Set in time 2O(dk)nO(1)2^{O(dk)}\cdot n^{O(1)} using nO(1)n^{O(1)} space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with dd if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

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    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^(dk) ⋅ n^(1) using (dk²log n) space; - Max Cut can be solved in time n^(dk) using (dk log n) space; and - Dominating Set can be solved in time 2^(dk) ⋅ n^(1) using n^(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial

    Colouring Complete Multipartite and Kneser-type Digraphs

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    The dichromatic number of a digraph DD is the smallest kk such that DD can be partitioned into kk acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations. Extending a well-known result of Lov\'{a}sz, we show that the dichromatic number of the Kneser graph KG(n,k)KG(n,k) is Θ(n2k+2)\Theta(n-2k+2) and that the dichromatic number of the Borsuk graph BG(n+1,a)BG(n+1,a) is n+2n+2 if aa is large enough. We then study the list version of the dichromatic number. We show that, for any ε>0\varepsilon>0 and 2kn1/2ε2\leq k\leq n^{1/2-\varepsilon}, the list dichromatic number of KG(n,k)KG(n,k) is Θ(nlnn)\Theta(n\ln n). This extends a recent result of Bulankina and Kupavskii on the list chromatic number of KG(n,k)KG(n,k), where the same behaviour was observed. We also show that for any ρ>3\rho>3, r2r\geq 2 and mlnρrm\geq\ln^{\rho}r, the list dichromatic number of the complete rr-partite graph with mm vertices in each part is Θ(rlnm)\Theta(r\ln m), extending a classical result of Alon. Finally, we give a directed analogue of Sabidussi's theorem on the chromatic number of graph products.Comment: 15 page

    On the sizes of generalized cactus graphs

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    A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a kk-cactus is a connected graph in which each edge is contained in at most kk cycles where k1k\ge 1. It is well known that every cactus with nn vertices has at most 32(n1)\lfloor\frac{3}{2}(n-1) \rfloor edges. Inspired by it, we attempt to establish analogous upper bounds for general kk-cactus graphs. In this paper, we first characterize kk-cactus graphs for 2k42\le k\le 4 based on the block decompositions. Subsequently, we give tight upper bounds on their sizes. Moreover, the corresponding extremal graphs are also characterized. However, the case of k5k\ge 5 remains open. For the case of 2-connectedness, the range of kk is expanded to all positive integers in our research. We prove that every 22-connected k (1)k ~(\ge 1)-cactus graphs with nn vertices has at most n+k1n+k-1 edges, and the bound is tight if nk+2n \ge k + 2. But, for n<k+1n < k+1, determining best bounds remains a mystery except for some small values of kk.Comment: 14 pages, 2 figure

    Structured Semidefinite Programming for Recovering Structured Preconditioners

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    We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite KRd×d\mathbf{K} \in \mathbb{R}^{d \times d} with nnz(K)\mathrm{nnz}(\mathbf{K}) nonzero entries, computes an ϵ\epsilon-optimal diagonal preconditioner in time O~(nnz(K)poly(κ,ϵ1))\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1})), where κ\kappa^\star is the optimal condition number of the rescaled matrix. We give an algorithm which, given MRd×d\mathbf{M} \in \mathbb{R}^{d \times d} that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in M\mathbf{M} in O~(d2)\widetilde{O}(d^2) time. Our diagonal preconditioning results improve state-of-the-art runtimes of Ω(d3.5)\Omega(d^{3.5}) attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of Ω(dω)\Omega(d^{\omega}) where ω>2.3\omega > 2.3 is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.Comment: Merge of arXiv:1812.06295 and arXiv:2008.0172

    The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

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    We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between ω(logn)\omega(\log^* n) and o(logn)o(\log n). This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where O(1)O(1), Θ(logn)\Theta(\log^* n), and Θ(n)\Theta(n) are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes
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